INVENTORY-PRODUCTION CONTROL SYSTEMS WITH GUMBEL DISTRIBUTED DETERIORATION

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1 Jurnal Karya Asli Lorekan Ahli Maemaik Vol. 6 No. (03) Page Jurnal Karya Asli Lorekan Ahli Maemaik INVENTORY-PRODUCTION CONTROL SYSTEMS WITH GUMBEL DISTRIBUTED DETERIORATION M. Azizul Baen School of Quaniaive Sciences, Universii Uara Malaysia, 060 UUM Sinok Keah Malaysia Anon Abulbasah Kamil an Habibah Laeh School of Disance Eucaion, Universii Sains Malaysia, 800 USM, Penang, Malaysia Yunus Ahme Khan 3 Deparmen of Geology an Mining, Universiy of Rajshahi, Rajshahi, Banglaesh Absrac : This paper is concerne wih he opimal conrol of invenory-proucion sysem subjec o Gumbel isribue eerioraion iems wih applicaions o opimal conrol heory. We successfully formulae he moel as a linear opimal conrol problem an obaine an explici soluion using Ponryagin maximum principle. The opimaliy coniions are erive in his case. I is hen illusrae wih he help of examples. Key Wors: Invenory-proucion moel, Opimal conrol, Perishable iems, Gumbel isribuion, Sensiiviy analysis. Mahemaics Subjec classificaions: 90B05, 93Cxx.. Inroucion We are concerne wih invenory-proucion conrol problem ha can be represene as an opimal conrol problem wih one sae variable (invenory level) an one conrol variable (rae of manufacuring) subjec o ime of eerioraion. In invenory-proucion moels, wo facors of he problem have been of growing ineres o he researchers, one being he eerioraion of iems an he oher being he variaion in he eman rae wih ime. We are especially inerese in he applicaion of opimal conrol heory o he proucion planning problem wih Gumbel isribuion eerioraion iems. The opimal conrol heory has been applie o ifferen invenory-proucion conrol problems where researchers are involve o analyze he effec of eerioraion an he variaions in he eman rae wih ime in logisics. In he early sage of he suy, mos of he eerioraion raes in he moels are consan, such as Ghare an Schraer (963), Shah an Jaiswal (977), Aggarwal (978), Pamanabhana an Vrab (995), an Bhunia an Maii (999). Many researchers have exensively suie in his area of invenory ino consieraion in proucion policy making wih eerioraing iems, such as Goyal an Gunasekaran (995), Jiang an Du (998), Gong an Wang (005), Maiy e al. (007) an so on. In his siuaion, here are several scenarios; incluing eerioraion rae is a linear increasing funcion of ime [Bhunia an Maii (998), Mukhopahyay e.al. (004)], eerioraing rae is wo-parameer Weibull isribue [Wee (999), Mahapara (005), Wu an Lee (003). Chen an Lin (003), Ghosh an Chauhuri (004), Al-khehairi an Taj (007) an Baen an Kamil (009)], eerioraing rae is hree-parameer Weibull isribue [Chakrabary e. al. (998)], an eerioraing rae is oher funcion of ime [Aba (00)]. Various auhors aacke heir research in he applicaion of opimal conrol heory o he proucion planning problem. Some of hem are: Sehi an Thompson (000), Salama (000), Rials 03 Jurnal Karya Asli Lorekan Ahli Maemaik Publishe by Pusaka Aman Press Sn. Bh.

2 Jurnal KALAM Vol. 6, No., Page an Benne (00), Zhang e. al. (00), Khemlnisky an Gerchak (00), Hejar e al. (004, 007), Bounkhel an Taj (005), an Awa El-Gohary e. al., (009). In his conex, Ponryagin maximum principle has been use o eermine he opimal proucion cos conrol by (Bounkhel an Taj, 005; Taj e al., 006; Benhai, Taj an Bounkhel, 008). In paricular, Srinivasa Rao e al. (005, 007) who suie he invenory moels wih Pareo isribuion eerioraion rae o erive opimal orer quaniy wih oal cos minimize. Bu no aemp has been mae o evelop he invenory moel as an opimal conrol problem an erive an explici soluion of an invenory moel wih generalize exreme value isribuion (e.g. Gumbel) eerioraion using Ponryagin maximum principle. The generalize exreme value isribuion (Gumbel, 958) has wie applicabiliy because i is base on he assumpion ha he ranom variable of ineres has a probabiliy isribuion whose righ ail is unboune an is of an exponenial ype, which inclues imporan probabiliy ensiy funcions such as normal, lognormal, an gamma probabiliy funcions. The coninuous review policy of opimal conrol approach is also o be novel in his framework. There seems o be no lieraure on he opimal conrol of coninuous review manufacuring sysems wih generalize exreme value i.e. Gumbel isribuion eerioraion iems rae. In he presen paper, we assume ha he eman rae is ime-epenence an he ime of eerioraion rae is assume o follow a Gumbel isribuion as well as a non-negaive iscoun rae is consiere for he invenory sysems. The novely here is ha he ime of eerioraion is a ranom variable followe by Gumbel isribuion an we consier he problem of conrolling he proucion rae of a coninuous review manufacuring sysem. This paper evelops an opimal conrol moels an uilizes Ponryagin maximum principle by Ponryagin e al. (96) o erive he necessary opimaliy coniions for invenory sysems This paper evelops a firs moel in which he ynamic eman is a funcion of ime an of he amoun of on han-sock. We hen exen his firs moel o an even more general moel in which iems eerioraion are aken ino accoun which refer o Gumbel isribuion correspons o an exreme value isribuion. The paper also erives explici opimal policies for he invenory moels where iems are eerioraing wih Gumbel isribuion ha can be use in he ecision making process. The res of he paper is organize as follows. Following his inroucion, secion iscusses Gumbel isribuion wih is applicaions an he associae eerioraion rae funcion. In secion 3 we explain he firs invenory-proucion moel wih necessary assumpions. The similar evelopmens are conuce for he secon an hir moels. Secion 4 evelops opimal conrol problems. In secion 5 we suy he opimal conrol of he sysem an erive explici soluion of he moels. In secion 6 he illusraive examples of he resuls are given. Finally he las secion conclues he paper.. Gumbel Disribuion an Deerioraion Rae Funcion The Gumbel isribuion has been he mos common probabilisic moel use in moeling hyrological exremes (Brusaer, 005). I is perhaps he mos wiely applie saisical isribuion which is known as he exreme value isribuion of ype I an is applicaion areas in gobal warming problem, lanslie moeling, floo frequency analysis, offshore moeling, rainfall moeling an win spee moeling. Various applicaions of his exreme value o problems in engineering, climaology, hyrology, an oher fiels were presene by Kimmison (985). A recen book by koz an Naarajah (000), which escribes his Gumbel isribuion, liss over 50 applicaions, ranging from accelerae life esing hrough o earhquakes, floos, win guss, horse racing, rainfall, queues in supermarkes, sea currens, win spees an rack race recors ec. An imporan assumpion for his exreme value isribuion (Gumbel, 958) is ha he original sochasic process mus consis of a collecion of hese ranom variables ha are inepenen an ienically isribue. In his aricle, we propose a generalize of he Gumbel isribuion wih he hope ha i will arac wier applicabiliy in invenory proucion cos conrol moeling where he novely we ake ino consieraion in his suy is ha he ime of eerioraion is a ranom variable followe by he

3 M. Azizul Baen e al. hree-parameer generalize exreme value isribuion. The probabiliy ensiy funcion of a generalize exreme value isribuion having probabiliy isribuion of he form () exp / f exp exp / where R 0,, is he shape parameer. The shape parameer governs he ail behavior of he isribuion. The family efine by 0 correspons o he Gumbel isribuion. The probabiliy ensiy funcion of Gumbel isribuion correspons o a minimum value is he locaion parameer an is he scale parameer an fmin ( ) exp / expexp /, 0, an he cumulaive isribuion funcion of Gumbel isribuion Fmin ( ) exp exp /, 0. The eerioraion assessmen is conuce primarily by scieniss an engineers. The resuls of such an invenory proucion eerioraion cos assessmen can be incorporae ino a socioeconomic framework o provie a sysem for evaluaing perinen social or economic risks, where he erm risk implies suscepibiliy o losses. The eerioraion rae funcion efine by f() (), 0is an imporan quaniy characerizing life phenomena. The insananeous F ( ) rae of eerioraion of Gumbel isribuion correspons o a minimum value of he on-han invenory is given by The firs erivaive wih respec o is f () min min () exp /, 0. Fmin ( ) Thus, () is an increasing funcion of. 3. The Moel 3. Moel wihou Iem Deerioraion min () exp /, 0. We are concerne wih he opimal conrol problem on inerval [0, T ] o minimize he iscoune cos conrol of proucion planning in an invenory sysem T 0 minimize Juxu,, e qx () x () ru () u?() () 3

4 Jurnal KALAM Vol. 6, No., Page subjec o he ynamics of he invenory level of he sae equaion which says ha he invenory a ime is increase by he proucion rae u () an ecrease by he eman rae y () can be wrien as accoring o x() [ u() y()] () xt an he non-negaiviy consrain wih iniial coniion ( ) 0 u ( ) 0, for all 0, T where he fixe lengh of he planning horizon is T, x(): invenory level funcion a any insan of ime [0, T], u (): proucion rae a any insan of ime [0, T] an y () : eman rae a any insan of ime [0, T], q : invenory holing cos incurre for he invenory level o eviae from is goal, r : proucion uni cos incurre for he proucion rae o eviae from is goal, xˆ( ) : invenory goal level, u ˆ( ): proucion goal rae, 0 : consan non-negaive iscoun rae. We wan o keep he invenory x() as close as possible o is goal xˆ( ), an also keep he proucion rae u () as close o is goal level u ˆ( ). The quaraic erms qx [ ( ) x ˆ( )] an ru [ ( ) u ˆ( )] impose 'penalies' for having eiher x or u no being close o is corresponing goal level. The curren-value Hamilonian of he problem is efine as ˆ T e qx x ru u u y H, x, u, u, () () () (). (3) 3. Moel wih Iem Deerioraion 0 Consier a sysem where iems are subjec o Gumbel isribue eerioraion correspons o a minimum value isribuion. For 0, le h () exp / be he eerioraion rae a he invenory level x() a ime. Keeping same noaion an he same opimal conrol problem as in he previous secion, he ynamics of he invenory level of he sae equaion which says ha he invenory a ime is increase by he proucion rae u () an ecrease by he eman rae y () an he rae of eerioraion exp / of Gumbel isribuion correspons o a minimum value isribuion can be wrien as accoring o x() [ u() y() exp / x()] (4) xt an he non-negaiviy consrain u T wih iniial coniion ( ) 0 The curren-value Hamilonian of he problem is efine as () 0, for all 0,. ˆ T H, x, u, u, e q x() x() r u() u() u y exp / x(). (5) Assumpions 4

5 M. Azizul Baen e al. Le us consier ha a manufacuring firm proucing a single prouc, selling some an socking he res in a warehouse. We assume ha an invenory goal level an a proucion goal rae are se, an penalies are incurre when he invenory level an he proucion rae eviae from hese goals. Again, we assume ha he proucion eerioraes while in sock an he eman rae varies wih ime. The firm has se an invenory goal level an proucion goal rae. Since he consrain u ( ) y 0, for all 0, T wih he sae equaion x is non-ecreasing. Therefore, shorages are no allowe in his suy. Finally assume ha he insananeous rae of eerioraion of he on-han invenory follows he hree parameers generalize exreme value isribuion an he proucion is coninuous. 4. Developmen of he Opimal Conrol Moel ha In orer o evelop he opimal conrol moel, efine he variables z(), z () an () such z () x () x (), (6) z () u () u (), (7) an ( ) u ( ) y( ) exp / x( ). (8) Aing an subracing he las erm exp / xˆ ( ) from he righ han sie of equaion (8) o he equaion (4) an rearranging he erms we have x() x () [ exp / ( x() x()) u() y() exp / x? ()]. Hence by (6) z( ) [ exp / z( ) u( ) y( ) exp / xˆ ( )]. (9) Now subsiuing (7) an (8) in (9) yiels z( ) [ exp / z( ) z ( ) ( )]. (0) The opimal conrol moel () becomes T minimize Jz, z e { q[ z( ) ] r[ z( ) ]} () 0 subjec o an orinary ifferenial equaion (0) an he non-negaiviy consrain z ( ) 0, for all 0, T. By he virue of () he insananeous sae of he invenory level () x a any ime is governe by he ifferenial equaion 5

6 Jurnal KALAM Vol. 6, No., Page x() u () y (), 0 T, xt ( ) 0 () The bounary coniions wih he equaion (3) are: a x xt (0) 0, 0 x () u () y, for 0 T. (3) Assuming ha x(0) x is known an noe ha he proucion goal rae u ˆ( ) can be compue using he sae equaion () as u ˆ( ) y () (4) By he virue of (4) he insananeous sae of he invenory level x() a any ime is governe by he ifferenial equaion x () exp / x () u () y (), 0 T, x ( T ) 0 (5) This is a linear orinary ifferenial equaion of firs orer an is inegraing facor is exp{ exp / } exp exp /. Muliplying boh sies of (6) by exp exp / T 0 an hen inegraing over [0, T ], we have x ( )exp exp / x(0) [ y ( ) u ( )]exp exp /, (6) Subsiuing his value of x (0) in (5), we obain he insananeous level of invenory a any ime [0, T] is given by x [ y ( ) u ( )]exp exp / [ y ( ) u ( )]exp exp / 0 0 (). exp exp / Solving he ifferenial equaion he on-han invenory a ime is obaine as x () x 0 exp exp / [ y () u ()] 0 T. (7) Assuming ha x(0) x is known an noe ha he proucion goal rae u ˆ( ) can be compue using he sae equaion (5) as u ( ) y () exp / x () (8) T 0 T 6

7 M. Azizul Baen e al. 5. Soluion o he Opimal Conrol Problem In orer o solve he opimal conrol problem () subjec o sae equaion () an (4), we erive he necessary opimaliy coniions using maximum principle of Ponryagin (96), see also Sehi an Thompson (000). 5. Soluion of he Opimal Conrol Problem wihou Iem Deerioraion The opimal conrol approach consiss in eermining he opimal conrol u ˆ( ) ha minimizes he opimal conrol problem () subjec o he sae equaion (). By he maximum principle of Ponryagin (96), here exiss ajoin funcion () such ha he Hamilonian funcional form (3) saisfies he conrol equaion he ajoin equaion an he sae equaion Hx, ( ), u ( ), u ˆ( ), ( ) 0, (9) u () Hx, (), u (), u ˆ(), () (), ( T) 0 (0) x () H, x(), u(), uˆ (), () x(), x(0) 0. () () Then he conrol equaion is equivalen o The ajoin equaion is equivalen o an he sae equaion is similar o (). e u () u ˆ() (). () r () qe x xˆ, (3) Subsiuion expression () ino he sae equaion () yiels () e x () u ˆ() y (). (4) r From which we have () e x () u ˆ() y (). (5) r 7

8 Jurnal KALAM Vol. 6, No., Page By iffereniaing (4), we obain x () u ˆ() y () e () () e (6) r An subsiuion expression (3) ino he equaion (6) yiels q x () u () y () x () x () e. (7) r r Finally, subsiuing expression (5) ino (7) o obain q q x () x () u () y () x () y () u?(). (8) r r Since a close form soluion is no possible, so his bounary value problem can be solve numerically ogeher wih iniial coniion x(0) 0 an he erminal coniion ( T ) Soluion of he Opimal Conrol Problem wih Iem Deerioraion The opimal conrol approach consiss in eermining he opimal conrol u ˆ( ) ha minimizes he opimal conrol problem () subjec o he sae equaion (4). By he maximum principle of Ponryagin (96), here exiss ajoin funcion () such ha he Hamilonian funcional form (5) saisfies he necessary coniions (9), (0) an (). Then here he conrol equaion (9) is equivalen o () also. The ajoin equaion () is equivalen o ( ) qe exp / qe xˆ, (9) An he sae equaion () is similar o (4). Subsiuion expression () ino he sae equaion (4) yiels () e x () u ˆ() y () exp / x (), (30) r from which we have () e x () u ˆ() y () exp / x (). (3) r By iffereniaing (30), we obain 8

9 M. Azizul Baen e al. ˆ x () u () y () e () () e exp / /. (3) r Subsiuion expression (9) ino he equaion (3) yiels q x () u () y () x () x () e x exp / r r exp / /. (33) Finally, subsiuing expression (3) ino (33) o obain q x() exp e / exp / x() r r q u () y () x () y () u?(). (34) r Since a close form soluion is no possible, so his bounary value problem can be solve numerically ogeher wih iniial coniion x(0) 0 an he erminal coniion ( T ) Illusraive Examples In his secion, we presen some numerical examples. Numerical examples are given for four ifferen cases of eman raes. a. Deman rae is consan: y () y 0, b. Deman rae is linear funcion of ime: y()=y ( ) y( ) 5, y() sin. c. Deman is sinusoial funcion of ime:. Deman is exponenial increasing funcion of ime: y () exp. In orer o presen illusraive examples of he resuls obaine we use he following parameers where he planning horizon has lengh T= monhs, 0.00, he invenory holing cos coefficien q 5 he proucion cos coefficien r 5. The goal invenory level is consiere xˆ( ) sin(), an he locaion an scale parameers of he Gumbel isribuion rae are consiere as an respecively. Then he eerioraion rae of Gumbel isribuion becomes h () exp, [0, T]. The invenory level x() in-erms of he firs-orer ifferenial equaion from (4) an he secon-orer ifferenial equaion (34) consiering he above eman funcions are solve numerically using he version 6.5 of he mahemaical package MATLAB. 6. Consan Deman Funcion In his subsecion, we presen he moel wih consan eman funcion. Subsiuing consan y ( ) y 0 insea of y () in he conrolle sysem (4) we have x () u() y() exp / x(), 0 T, x( T) 0 9

10 Jurnal KALAM Vol. 6, No., Page from which he proucion goal rae u ˆ( ) can be compue (assuming x(0) u( ) y() exp / x(). x ) as Figure : The invenory level x() in-erms of he firs-orer ifferenial equaion in erms of consan eman. The consan eman rae is assume o have fixe value 0 unis per uni ime. Noe ha here eman an eerioraion ecrease he invenory level isplaye in Figure. From he Figure, i is clear ha he proucion rae is no following he consan eman rae bu he proucion rae wih consan eman increases over ime. Figure : Opimal Proucion Policy wih Consan Deman. 6. Linear Deman Funcion In his subsecion, we presen he moel wih linear eman funcion. Subsiuing linear y ()= 5 insea of y () in he conrolle sysem (4) we have 0

11 M. Azizul Baen e al. x() u() y() exp / x(), 0 T, x( T) 0 from which he proucion goal rae u ˆ( ) can be compue (assuming x(0) x ) as u( ) y() exp / x(). Figure 3: The invenory level x() in-erms of he firs-orer ifferenial equaion in erms of linear eman. In case of linear eman, i is he form y()= 5an he invenory level in-erms of firsorer ifferenial equaion ecreases over ime shown in Figure 3. The resul is shown in Figure 4 an i is foun ha he proucion rae is no following he linear eman rae. The proucion rae sars wih zero amoun an increases over ime. Figure 4: Opimal Proucion Policy wih Linear Deman. 6.3 Sinusoial Deman Funcion

12 Jurnal KALAM Vol. 6, No., Page In his subsecion, we presen he moel wih sinusoial eman funcion. Subsiuing y3( ) sin() insea of y () in he conrolle sysem (4) we have x3() u3() y3() exp / x3(), 0 T, x( T) 0 from which he proucion goal rae u ˆ( ) can be compue (assuming x(0) x ) as u( 3 ) y3() exp / x3(). Figure 5: The invenory level x() in-erms of he firs-orer ifferenial equaion in erms of sinusoial eman. Figure 6: Opimal Proucion Policy wih Sinusoial Deman. In Figures 5 o 8 o no show he variaions of he invenory an opimal proucion level wih ime wih changing he shape of he eman funcions. In case of sinusoial an exponenial

13 M. Azizul Baen e al. ecreasing eman oriene opimal invenory levels over ime almos have no variaions ha suppor he finings of Baen an Kamil (009). I is observe ha he opimal proucion raes are no very sensiive o changes in he eman funcions in case of Gumbel isribuion. 6.4 Exponenial Increasing Deman Funcion In his subsecion, we presen he moel wih sinusoial eman funcion. Subsiuing y 6 () exp insea of y () in he conrolle sysem (4) we have x6 () u6() y6() exp / x6(), 0 T, x( T) 0 from which he proucion goal rae u ˆ( ) can be compue (assuming x(0) x ) as u( 6 ) y6() exp / x6(). Figure 7: The invenory level x() in-erms of he firs-orer ifferenial equaion in erms of exponenial increasing eman. Figure 8: Opimal Proucion Policy wih Exponenial Increasing Deman. 3

14 Jurnal KALAM Vol. 6, No., Page Figure 9: The invenory level x() in-erms of he secon-orer ifferenial equaion. The soluion of he secon-orer ifferenial equaion is represene by Figure-9 an shows he sae of opimal invenory level is increasing. However, in he subsecions we presen he moel o measure he performance using ifferen eman paerns. The proucion level wih ime given u ˆ( ) from he equaion (8) consiering he menione above ifferen eman raes an we ake he invenory goal level is as x ˆ( ) 0keeping all oher parameers unchange. 7. Conclusions In his paper, we evelope an opimal conrol moel in invenory-proucion sysem wih generalize Gumbel isribuion eerioraing iems. This paper erive he explici soluion of he opimal conrol moels of an invenory-proucion sysem uner a coninuous review-policy using Ponryagin maximum principle. However, we gave numerical illusraive examples for his opimal conrol of a proucion-invenory sysem wih Gumbel isribuion eerioraing iems. However, paricular emphasis can be mae also wih exreme value (e.g. Gumbel) isribuion for assessing lanslies hazars, earhquake hazars. Near-surface grounwaer levels can be reae he same exreme-value isribuion conex as river floos, win guss, earhquakes, an oher naurally occurring emporal phenomena. References Aba, P. L. 00. Opimal price an orer size for a reseller uner parial backorering. Com. & Opera. Res.. 8, Aggarwal, S. P A noe on an orer-level invenory moel for a sysem wih consan rae of eerioraion. Opsearch. 5, Awa El-Gohary e al., 009. Using opimal conrol o ajus he proucion rae of a eerioraing invenory sysem. J. of Taliban Uni.for Sci., Al-khehairi, A., L. Taj Opimal conrol of a proucion invenory sysem wih Weibull isribue eerioraion. Appl. Mahe. Sci. (35), Baen, M.A., A. A. Kamil An opimal conrol approach o invenory-proucion sysems wih Weibull isribue eerioraion. J. of Mahe. an Sa. 5(3) Bhunia, A. K., M. Maii An invenory moel of eerioraing iems wih lo-size epenen replenishmen cos an a linear ren in eman. App. Mahe. Moel. 3,

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